Politicians played hot potato with the No. 2 spot on the Republican presidential ticket on Sunday, with the latest round of prominent Republicans closing the door at varying degrees to serving as presumptive GOP presidential nominee Mitt Romney's vice president.

While the Republican Party lines up behind Romney after divisions emerged in the primaries, the chatter has increasingly turned to the man or woman the former Massachusetts governor will tap to run alongside him during the general election.

Sen. Marco Rubio of Florida was elected in 2010 aided by a wave of tea party support and a platform of fiscal conservatism. Speculation about his future has dogged the freshman representative since then, something the impending release of his memoir has done little to temper.

Rubio, the son of Cuban immigrants, has become one of Romney's most ardent backers of late and is scheduled to campaign with Romney outside Philadelphia on Monday.

He has consistently said he has no interest in the vice presidency and plans to focus on his Sunshine State constituents, but he adapted a new line on Sunday, vowing to stay out of the punditry.

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"I'm not even going to discuss the process anymore," Rubio told CNN Chief Political Correspondent Candy Crowley on CNN's "State of the Union." "The last thing he needs are those of us in the peanut gallery to be saying what we would or would not do."

When Crowley said his response would be "translated that you've backed off saying 'I would not accept it,'" Rubio said he wanted to be respectful of the process, now that it is officially under way.

Romney appointed Beth Myers last week to head his search for the individual who will complete the GOP ticket heading into the November general election.

Rubio is the latest in a series of supporters to appear under the VP spotlight around an event or series of events with Romney.

Joint appearances with Rep. Paul Ryan of Wisconsin, Gov. Nikki Haley of South Carolina, Gov. Chris Christie of New Jersey and Sen. Kelly Ayotte of New Hampshire and others have sparked similar chatter.

But the focus on Rubio has always been sharper, given his youth, popularity among many that associate with the tea party movement, battleground state roots and his place in the Hispanic community — a crucial voting bloc in which Romney trails President Barack Obama badly in most polls.

In pushing back against the suggestion he might be No. 2 on the GOP ticket, Rubio offered up former Florida Gov. Jeb Bush, saying he would make a "fantastic vice president."

Bush appeared to open the door to the position in an interview last week, but later told Bloomberg, "I am not going to be the veep nominee. Lay that to rest. I guess I wasn't clear enough."

Each of the potentials appears to have a slightly different way of saying "no," something they have happily or unhappily been forced to hone given the at times relentless questioning surrounding the choice.

Indiana Gov. Mitch Daniels, who decided against a White House bid of his own earlier in the year, went so far as to say he would "demand reconsideration" if asked to assume the position.

"I think I would demand reconsideration and send Mr. Romney a list of people I think could suit better," Daniels said on "Fox News Sunday."

The two-term governor and former George W. Bush budget chief would not offer any other suggestions, only saying "there is a lot of talent in the Republican Party."

A CNN/ORC International poll last week that gauged support for potential running mates proved the importance of name recognition early on in the process.

Condoleezza Rice, former President George W. Bush's secretary of state, received the highest level of support with 26%, followed by former presidential candidate and Pennsylvania Sen. Rick Santorum with 21%, Christie and Rubio tied at 14%, Ryan at 8%, Gov. Bobby Jindal of Louisiana at 5% and Gov. Bob McDonnell of Virginia at 1%. Sen. Rob Portman of Ohio received less that one half of one percent.

However, the survey indicated there is no consensus among Republicans, only that an ideological split existed, which could put pressure on Romney as he makes his decision. Self-identified tea party Republicans put Rubio and Christie at the top of their wish lists, followed by Rice and Santorum. But Christie and Rubio received little support among those who do not support the grassroots movement, once again in large part due to name recognition.

Portman, who received the lowest level of support in the recent poll, is thought to be an increasingly likely choice given he hails from the ever important Ohio and carries an impressive resume that includes serving as George W. Bush's director of the Office of Management and Budget, U.S. trade representative under Bush and a former congressman.

Although he did not offer a definitive "no" when asked by CNN Chief National Correspondent John King if he would join the GOP ticket in the next election, he opined that Rubio and Ryan have a better chance.

Ultimately, Portman said the vice presidential pick will hold little clout in November.

"Frankly, people vote for the person at the top of the ticket," Portman said Thursday on CNN's "John King, USA."

Portman certainly seems to have a point given that the candidate leading the ticket often does not make his selection known to the public until the final stretch of the campaign, around the convention.

In 2008, the choices of then candidates John McCain and Obama were announced a mere days before their respective conventions and were intended to fill perceived holes at the top of the ticket.

Then-Gov. Sarah Palin of Alaska helped rile up the Republican base, giving a much-needed boost to McCain coming out of his convention. Then-Sen. Joe Biden of Delaware lent Obama the foreign policy credentials that he lacked and more experience in how to navigate Congress.

CNN Senior Political Analyst Ron Brownstein said the current VP conversation seems early compared to previous campaigns, but that floating names is a way for the Romney campaign to reach different constituencies within the party and "generate interest among more casual voters."

"It probably reflects the need to develop and sustain a positive storyline for the Romney campaign that can help them drive the media debate in this long period, this kind of extended pre-season, before the general election begins in earnest," said Brownstein, editorial director for National Journal.

Although Romney said last week a short list does not yet exist, he guaranteed a thorough review of the candidates.

"We're beginning that process, we'll talk about a lot of folks, and then go through the kind of vetting and review process that you have to go through to make sure whoever you select will pass the evaluation that's required by the American people," Romney told ABC News.

Former Gov. John Sununu of New Hampshire, a longtime Romney supporter, bluntly said his candidate should pick a "dull choice," in the model of former picks like Walter Mondale, George H.W. Bush, Al Gore, Dick Cheney and Joe Biden.

"The winning choice is the dull choice -- a running mate the public already knows, warts and all," Sununu wrote recently in The Dallas Morning News, before referencing the names above. "These were not picks that lit the world on fire. They were serious, experienced names ... They weren't from key states and weren't part of some grand plan to balance ideology. But they all won."